Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Comparative Statics of Trading Boundary in Finite Horizon Portfolio Selection with Proportional Transaction Costs

Jintao Li, Shuaijie Qian

Abstract

We consider Merton's problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries, the buy boundary and the sell boundary, between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.

Comparative Statics of Trading Boundary in Finite Horizon Portfolio Selection with Proportional Transaction Costs

Abstract

We consider Merton's problem with proportional transaction costs. It is well known that the optimal investment strategy is characterized by two trading boundaries, the buy boundary and the sell boundary, between which lies the no-trading region. We investigate how these two trading boundaries vary with the transaction cost rates. We show that the cost-adjusted trading boundaries are monotone in the transaction costs. Our result implies the following: (i) the Merton line must lie between the two cost-adjusted trading boundaries; and (ii) when the Merton line is positive, both the buy and sell boundaries are monotone in the transaction cost rates, and consequently the Merton line lies in the no-trading region.

Paper Structure

This paper contains 19 sections, 12 theorems, 93 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Model setup
  3. The market
  4. The investor's problem
  5. Value function and HJB equation
  6. Trading and no-trading regions
  7. Impact of transaction cost rates
  8. Monotonicity in $\lambda$ and $\mu$
  9. Sell boundary when $\theta := \frac{1+\lambda}{1-\mu}$ is large
  10. Proof of Theorem \ref{['thm: transaction con monotone']}
  11. Numerical results
  12. Trading boundaries with respect to time
  13. Trading boundaries with respect to transaction cost rates
  14. Conclusion
  15. Proof of Lemma \ref{['prop compare transa cost']}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $x_s(t; \lambda,\mu)$ and $x_b(t; \lambda,\mu)$ denote the sell and buy boundaries of the finite-horizon problem equ investors problem with transaction cost rates $(\lambda,\mu)$. Then: (i) The sell boundary $x_s(t;\lambda,\mu)$ is monotonically decreasing in the purchase cost $\lambda$, and the

Figures (12)

  • Figure 1: Trading boundaries against time. Parameters: $T=2$, $\alpha = 0.3$, $r = 0.01$, and $\sigma = 0.2$. The corresponding Merton line is $x_M = -0.862$.
  • Figure 2: Trading boundaries (left) and cost-adjusted trading boundaries (right) for $\mu=\lambda$. Default parameters: $T=2$, $t=0.25$, $\alpha=0.3$, $r=0.01$, and $\sigma=0.2$. The corresponding Merton line is $x_M=-0.862$.
  • Figure 3: Trading boundaries (left) and cost-adjusted trading boundaries (right) against the sale cost rate $\mu$. Default parameters: $T=2$, $t=0.25$, $\alpha=0.3$, $r=0.01$, and $\sigma=0.2$. The corresponding Merton line is $x_M=-0.862$.
  • Figure 4: Trading boundaries (left) and cost-adjusted trading boundaries (right) against the purchase cost rate $\lambda$. Default parameters: $T=2$, $t=0.25$, $\alpha=0.3$, $r=0.01$, and $\sigma=0.2$. The corresponding Merton line is $x_M=-0.862$.
  • Figure 5: The critical value $\mu^*(t)$. Default parameters: $T=2$, $\alpha=0.3$, $r=0.01$, and $\sigma=0.2$.
  • ...and 7 more figures

Theorems & Definitions (19)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Corollary 1
  • Corollary 2
  • Remark 3
  • Corollary 3
  • Corollary 4
  • Theorem 2
  • Remark 4
  • ...and 9 more